Given two such extended matrices \(M_1\)  and \(M_2\) , a general algorithm to decide whether each regular category with \(M_1\) -closed relations has \(M_2\) -closed relations, denoted as \(M_1\Rightarrow _{\mathsf {reg}} M_2\) , is yet to be found. However, using the embedding theorems from [1]}, [2]}, the statement \(M_1\Rightarrow _{\mathsf {reg}} M_2\)  is equivalent (assuming the axiom of universes [3]}) to the statement, denoted by \(M_1\Rightarrow _{\mathsf {reg\, ess\, alg}} M_2\) , that any regular essentially algebraic category (in the sense of [4]}, [5]}) with \(M_1\) -closed relations has \(M_2\) -closed relations. Using this equivalence and the results from [6]}, we could prove that, when \(M_2=\mathsf {Mal}\) , the statement \(M_1\Rightarrow _{\mathsf {reg}} \mathsf {Mal}\)  is equivalent to the statement \(M_1\Rightarrow _{\mathsf {alg}} \mathsf {Mal}\) .
